Industrial Robotics (Theory, Modelling and Control) - P3 pot

6 Inverse Position Procedure for Manipulators with Rotary Joints Ibrahim A.. Another geometric aspect that leads to simplified inverse solutions is the spherical wrist design, which ent

Structure Based Classification and Kinematic Analysis of … 183

Appendix B

List of Six-joint Industrial Robots Surveyed (Balkan et al., 2001)

6

Inverse Position Procedure for Manipulators

with Rotary Joints

Ibrahim A Sultan

1 Introduction

Industrial robot manipulators are essentially spatial linkages that consist of rigid bodies connected by joints Even though many types of joints (which are also known as kinematic pairs) are available for use in mechanical linkages, only two types are employed for robot manipulators These are the revolute,

or rotary, joints (referred to in literature as R) and the prismatic, or sliding, joints (referred to as P) These specific types allow a single degree of freedom relative movement between adjacent bodies; and are easier to drive and con-trol than other kinematic pairs Normally every joint on the manipulator is in-dependently driven by a dedicated motor It is central to kinematic control of manipulators to calculate the sets of joint-motor displacements which corre-spond to a desired pose (i.e position and orientation) at the end-effector The mathematical procedure which is followed to achieve this purpose is often re-

ferred to as, Inverse Position Analysis This analysis presents a special difficulty

in the field of Robotics as it is associated with the use of intricate spatial ometry techniques The complexity of the analysis increases substantially with the number of rotary joints on the manipulator structure For this reason a considerable part of the published literature is mainly concerned with the revolute-joint manipulators

ge-Published literature reveals that various methods have been proposed to solve the inverse position problem of manipulators These methods range from Jacobian-based iterative techniques to highly sophisticated levels of equation-manipulation intended to reduce the whole model into a polynomial with thousands of mathematical terms However, most industrial robots are de-signed with geometric features (such as parallelism and perpendicularity) to make it possible for simple inverse position solutions to be obtained in closed forms suitable for real time control Another geometric aspect that leads to simplified inverse solutions is the spherical wrist design, which entails that the last three joints on the manipulator structure intersect at one point This usu-ally suggests that these three joints (also known as the wrist joints) have the main task of orienting (rather than placing) the end-effector in space In this

case it should be possible to regard the manipulator as consisting of two rate parts where the first part (referred to as the arm) consists of the first three joints, counting from the stationary base, on the structure The task of the arm

sepa-is to place the end-effector origin (i.e the point of intersection of the last three joint axes) at a defined point in space The solution for this first part can be ob-tained separately before proceeding to find the angles of the last three joints which will result in giving the end-effector its desired spatial orientation The work presented in this paper adopts this strategy to propose a mathematical procedure, for the arm inverse solution, based on assigning local coordinates

at every joint, and utilising the properties of rotation to relate these nates A model manipulation technique is then employed to obtain the arm inverse solution in terms of one polynomial A kinematic synthesis discussion

coordi-is then presented for the arm structure in terms of local coordinates to reflect

on the number of solutions expected from the polynomial It will be shown that the concept of intersecting spatial circles offers a good ground to compre-hend the kinematics of revolute-joint manipulators Moreover, models are presented for the wrist structure to obtain a full inverse kinematic solution for the robot manipulator A solved example is demonstrated to prove the valid-ity of the method presented

2 Literature Survey

Published literature reveals that the homogeneous transformation matrix which was developed as far back as 1955 has extensively been employed for the analysis of robot manipulators The matrix involves the use of four pa-rameters, usually referred to as the DH-parameters, intended to perform trans-formation between two spatial Cartesian coordinate systems (Denavit and Hartenberg, 1955) Recently, other kinematic models have been proposed by researchers to deal with the drawbacks of the DH presentation (Sultan and Wager, 1999) This is particularly important if the model is going to be imple-mented for robot calibration purposes The theory of dual-number algebra was introduced into the field of kinematics back in the 1960’s (Yang and Freu-denstein, 1964); and it did appeal to researchers in the field of robot kinematics

(Pennock and Yang, 1985; Gu and Luh, 1987; Pardeep et al, 1989) In addition

to these approaches, which are based on matrices, vector methods were also employed in the field of kinematic analysis of robots (Duffy, 1980; Lee and Li-ang, 1988A and 1988B)

Many industrial robots possess parallel and intersecting joint-axes and their direct-position models can be inverted analytically such that closed-form solu-tions may be obtained for the joint-displacements (Gupta, 1984; Pennock and

Yang, 1985; Pardeep et al, 1989; Wang and Bjorke, 1989)

Inverse Position Procedure for Manipulators with Rotary Joints 187 Spherical-wrist manipulators have their last three joint-axes intersecting at a common point For these manipulators the position of the end-effector in space is determined only by the displacements performed about the first three joint-axes This concept is often referred to as the position-orientation decoup-ling; and has been utilised to produce a closed form solution, for the inverse position problem of simple structure robots, efficient enough to be imple-mented for computer control (Pieper and Roth, 1969) Inverse position tech-niques have been proposed to utilise the position-orientation decoupling of industrial robot of arbitrarily directed axes (Sultan, 2000; Sultan and Wager, 2001) As such these techniques do not rely on any particular spatial relations (e.g parallelism or perpendicularity) between the successive joint-axes In fact, approaches which utilise these particular geometric features to produce the model equations are likely to produce positioning errors when used for ro-bot control since the actual structures always deviate from their intended ideal geometry.

Iterative techniques have been employed for the inverse position analysis of general robot manipulators Many of these techniques involve the computa-tion of a Jacobian matrix which has to be calculated and inverted at every it-eration The solution in this case may be obtained by a Newton-Raphson tech-nique (Hayati and Reston, 1986) or a Kalman filter approach (Coelho and Nunes, 1986) However, the inversion of the system Jacobian may not be pos-sible near singular configurations (where the motion performed about one joint-axis produces exactly the same effect, at the end-effector, as the motion performed about another axis, hence resulting in loss of one or more degrees

of freedom) Therefore, a singularity avoidance approach has been reported where the technique of damped least-squares is used for the analysis (Chia-

verini et al, 1994) However, this technique seems to be rather sluggish near

singular points where extra computational procedure may have to be volved

in-Optimisation techniques have also been employed to solve the position problem of manipulators whereby a six-element error vector was im-

inverse-plemented for the analysis (Goldenberg et al, 1985) The vector combines the

current spatial information (position and orientation) of the robot hand and compares it to the desired pose to produce error values Published literature

in the area of optimisation report a technique by which the robot is moved about one joint at a time to close an error gap (Mahalingam and Sharan, 1987; Wang and Chen, 1991; Poon and Lawrence, 1988) More recent research effort

demonstrates valuable inputs form such areas as neural networks (Zhang et al, 2005) and fuzzy techniques (Her et al, 2002) to the field of robot inverse kine-

matics

It has been shown that the kinematic behaviour of robots can be described in terms of a set of polynomials that can be solved iteratively (Manseur and Doty 1992a, 1992b and 1996) One such method features a set of eight polynomials

which were solved numerically to obtain different possible solutions to the

in-verse position problem; it could therefore be concluded that the maximum

number of meaningful solutions to the inverse position problem of a general

robotic structure is 16 (Tsai and Morgan 1985), rather than 32 as had

previ-ously been suggested (Duffy and Crane, 1980) However it has been pointed

out that a manipulator with 16 different real inverse position solutions can

sel-dom be found in real life (Manseur and Doty, 1989) In reality most

manipula-tors are designed to possess up to 8 solutions of which only one or two can be

physically attained

It is possible to express the inverse position problem of robots in terms of a 16

degree polynomial in the tan-half-angle of a joint-displacement (Lee and Liang

1988a & 1988b; Raghavan and Roth 1989) However it has been argued that

the coefficients of such a polynomial are likely to contain too many terms

which may render such a tack impractical to use (Smith and Lipkin 1990)

Also, these high order polynomials are obtained by evaluating the eliminants

of hyper-intricate determinants which may be impossible to handle

symboli-cally in the first place This may have motivated some researchers (Manocha

and Canny 1992; Kohli and Osvatic 1993) to reformulate the solutions in terms

of eigenvalue models in order to simplify the analysis and avoid numerical

complications However, a numerical technique has been introduced to obtain

the inverse solutions without having to expand the system characteristic

de-terminant (Sultan, 2002)

The procedure introduced here for the inverse position analysis of robot

ma-nipulators is described in the rest of this paper

3 Rotation of Vectors

The unit vector ˆziin Figure (1) represents an axis of rotation in a spatial

mechanism It is required to obtain the new rotated vector, vir, which results

from rotating the original vector vio (where vio× ≠zˆi 0) by an angle θizˆi In

or-der to do so, the Cartesian system ˆ ˆ ˆx y zi i i may be introduced as follows,

The original vector, vio, and the rotated vector vir,can both be expressed with

respect to the ˆ ˆ ˆx y z -frame in terms of local coordinates, n, m and l as follows,

Inverse Position Procedure for Manipulators with Rotary Joints 189

Figure 1 Rotation of Vectors

where the local coordinates are given as follows;

The inverse of this problem is encountered when ˆzi , vio and vir are all known

and it is required to obtain the corresponding value of θi With the values of

the local coordinates known, θicould be obtained as follows,

i m n ir io n m n n ir io ir io m m ir io

where the function atan2(y,x) is available in many computer algebra packages and compilers to compute the angle θi (over the range of the whole circle) when its sine and cosine are both given In this paper, the concepts mentioned above are used together with the suitable conditions of rotation to perform the inverse position analysis of the manipulator arm and wrist The proposed analysis for the arm is given in the next section.

4 Inverse Kinematics of the Arm

The arm, which is the largest kinematic part of the manipulator, consists of three revolute joints connected through rigid links Each joint, as shown in Figure (2), is represented by the spatial pose of its axis The first joint-axis has

a fixed location and orientation in space as it represents the connection tween the whole manipulator and the fixed frame Any other joint-axis num-

be-ber i can float in space as it rotates about the joint-axis numbe-ber i–1.

In the current context, the main function of the arm is to displace a certain tial point from an initial known location to a required final position In spheri-cal-wrist manipulators, this point is at the intersection of the wrist axes In a calibrated (non-spherical-wrist) manipulator, it may represent a point on the sixth axis as close as possible to the fifth joint-axis In Figure (2), the arm is re-quired to displace point pi to a final position pf The position vectors, pib and

tance, a i+1, between the axes, ˆzi and zˆi+1, is measured along xˆi+1 which sects ˆzi at the point pi and zˆi+1 at the point pi i(+1).

inter-At the zero initial position which is shown in Figure (2), the axis ˆx is chosen 1

to coincide with ˆx In this figure, the position vectors, 2 pi3o and pf1r, of points pi and pf respectively with respect to the frames x y zˆ ˆ ˆ3 3 3 and x y zˆ ˆ ˆ1 1 1 may

be numerically calculated as follows,

where p1 and p32 are the position vectors of the axes-attached, points p1 and

p , respectively as measured from the origin of the base coordinates

Accord-Inverse Position Procedure for Manipulators with Rotary Joints 191

ing to the concepts in (4) and (5), pf1r can be described with respect to the

p21

pi pf

Figure 2 A General View of a 3R Manipulator Arm at Its Zero Position

It is understood that the vector pf1r resulted from rotating another vector pf1o

about the ˆzi axis by an angle, θ1 (i.e a θ1 1ˆz -type rotation) The original vector,

1o

pf , can be expressed with respect to the x y zˆ ˆ ˆ1 1 1-framein terms of local

coor-dinates (n 1o, m 1o and l 1o) Also, during the positioning process the vector pi3o

will perform a θ3ˆz -type rotation to evolve into 3 pi3r which can be expressed

with respect to the x y zˆ ˆ ˆ3 3 3-frame in terms of local coordinates (n , 3r m and 3r

where the two equations above have four unknowns that need to be

deter-mined These four unknowns are n 1o, m 1o, n 3r and m 3r The numerical values

of the l-type local coordinates are calculated as follows;

In fact the value of l is calculated, and stored in a data file, once the manipu- 3r

lator has been calibrated and an initial position has been nominated; however,

1o

l has to be calculated for every new desired end-effector position Moreover,

the end-effector positions which are defined by the vectors pf1o and pi3r can

be used to study the rotation about the middle joint-axis, ˆz2 These same

posi-tions can be expressed relative to a point, p , attached to 2 ˆz2, using the two

re-spective vectors, p2r and p20 as follows;

It may be noted that p2r and p20 are separated by a single rotation, θ2ˆz2 The

properties of this rotation may be utilised to show that,

The concept in equations (3) and (5) may be employed to express the x2-,

y2-and z2-components of a rotated vector 2

r o

p which results from performing a

θ2ˆz rotation on 2 p20 Then the coincidence of p2r o and p2r may be described

by,

Inverse Position Procedure for Manipulators with Rotary Joints 193

p z p z is already described in equation (13) , and the

ex-panded forms of the (15) and (16) are given respectively as follows;

where c2 and s2 stand for cosθ2 and sinθ2 respectively

The four linear equations, (13), (14), (17) and (18) represent the mathematical

core of the kinematic model introduced in the present work for the inverse

po-sition analysis of the arm module A symbolic solution for these equations can

be obtained such that, n 3r and m 3r are expressed in the following forms,

wheref f1 and f2 are linear functions of s2 and c2

Noting the properties of rotation about ˆz3 the following may be deduced,

c and s c2 2; and can be expressed in the following form,

where the coefficients are calculated symbolically from the model presented

above The parameters which constitute these coefficients reflect the kinematic

relations between the successive arm axes and they can be calculated and

stored for run-time use once the arm is calibrated These parameters are all

constant for a given arm except pf1rDpf1r and l 1o which depend on the desired

final location of the end-effector as described above The fact that only two

pa-rameters need to be calculated highlights the computational efficiency of the

Equation (22), which is a fourth degree polynomial, can be solved using a

sys-tematic non-iterative technique (Tranter, 1980) The resulting roots can

succes-sively be plugged back in equations (23) to work out the corresponding values

of c2 and s2 These values are then employed to obtain the joint-displacement

2

θ using the atan2 function referred to above The values of the local

coordi-nates, n 3r and m 3r, may be calculated by using equations (19) and (20)

A numerically stable method to obtain m 1o and n 1o is to use equation (17) for

Finally, n 3r, m 3r, n 3o, m 3o are employed in equation (6) to obtain the

corre-sponding values of θ3 Similarly, n 1r, m 1r, n 1o, m 1o are used to obtain the

cor-responding values of θ1

As revealed by the polynomial in (22), the maximum number of arm

configu-rations, N arm, which correspond to a given end-effector position is four In

some cases, however, the geometrical relationships between the consecutive

axes as well as the required position of pf allow for the inverse position

prob-lem to be solved through the use of quadratic, rather quartic or higher,

poly-nomials Arms which exhibit this sort of simplification are said to have simple

structures Some of these cases are outlined in the next section

Inverse Position Procedure for Manipulators with Rotary Joints 195

5 Kinematic Synthesis of the Arm Mechanism

Most industrial robots are designed to have their successive axes either lel or perpendicular to make a simplified closed form inverse position solution

paral-achievable Researchers have repeatedly assigned the term, simple structure, to these robotic arms The word "simple" usually implies that a non-iterative so-

lution can be obtained for the inverse position problem of this particular ture However, as the discussion in the previous section reveals, a non-iterative solution can still be obtained even for arms with arbitrarily positioned and directed joint-axes A definition has been proposed for this term in the light of the conics theory (Smith and Lipkin, 1990) In the present section, a consistent simplified geometrical definition is introduced

struc-To gain understanding of the results obtainable from the fourth-degree nomial equation (22), equations (13) and (14) along with the following two equations may be considered,

Essentially, the inverse position problem of the arm structure may be depicted

as shown in Figure (3) In the figure, points pf and pi assume their local lar paths about the rotary axes, ˆz and 1 ˆz3, creating two spatial circles Cz1 and Cz3, respectively, in two planes perpendicular to ˆz and1 ˆz3 with their centres located on the axes Thus, a solution exists if a circle, Cz2, that intersects both Cz1 and Cz3 simultaneously, can be drawn in a plane normal to ˆz with its 2

circu-centre located along it As the analysis given in the previous section suggests,

if the three axes are located and directed arbitrarily in space, a maximum of four different circles can be drawn about ˆz to satisfy this condition Each cir-2cle intersects each of Cz1 and Cz3 at one point and hence, the four possible so-

lutions

Figure 3 The Kinematic Behaviour of the Arm Joints

As established in Appendix B, any two spatial circles may intersect at two points if, and only if, their corresponding axes lie in one and the same plane (this is the plane which perpendicularly halves the line connecting the two points of intersection) Therefore, in arms with ideal (non-calibrated) struc-tures where ˆz lies in the same plane with either 2 ˆz or 1 ˆz3, the number of

middle circles, Cz2, becomes two In such a case, the complex mathematical

aspects associated with the inverse position problem of the arm disappear and the solution can easily be obtained by using equations (13), (14), (25) and (26) For example, if ˆz and 1 ˆz lie in one plane where 2 a2 =0, the following proce-dure may be adopted for the solution;

- i use equations (13) and (14) to express n 3r and m 3r as functions of m 1o

- ii use these functions in equation (25) to obtain the two roots of m 1o: m1o1

Inverse Position Procedure for Manipulators with Rotary Joints 197

- iv use equations (13) to obtain the two corresponding values of m 3r:

The above mathematical procedure can be performed symbolically such that, closed form expressions are obtained for the three joint-displacements

Similar simplified mathematical procedure may be used in cases with ˆz par-1

allel to ˆz2 It may be noted that in designs where ˆz lies in one plane with 2 ˆz1

and in another plane with ˆz3, the number of middle circles, Cz2, becomes one

and the solution can be simplified even further In such a case, the middle cle, Cz2, intersects both Cz1 and Cz3 at two points to produce the four possible solutions An example may be sought in PUMA-type robots, whose nominal structures possess the following kinematic features,

- i obtain m 1o from equation (13)

- ii obtain n 3r from equation (14)

- iii obtain ±m 3r from equation (25)

- iv obtain ±n 1o from equation (26)

Thus, the four possible configurations of the arm are given by the following root combinations,

(n 1o, m 1o, n 3r, m 3r), (n 1o, m 1o, n 3r, −m 3r), (−n 1o, m 1o, n 3r, m 3r) and (−n 1o, m 1o,

3r

n , −m 3r)

Based on the above discussion it can be concluded that the middle axes ˆz2

must lie in one plane with either ˆz or 1 ˆz3 for a simplified mathematical cedure to be realisable Once this condition is satisfied, the four equations, (13), (14), (25) and (26) can be readily employed to obtain the inverse solution and therefore the arm structure can be described as simple

pro-In the next section, the procedure which is presented for the inverse position analysis of the wrist substructure is explained

6 Inverse Kinematics of the Wrist

In the current context, the main task of the first two wrist joints (namely the fourth and fifth joints on the manipulator structure) is to displace the axis of the last joint (i.e the sixth joint) from a given known orientation to a new de-sired direction in space

Figure (4) depicts an arrangement of two revolute joints with their axes ˆz and 4

coordi-Inverse Position Procedure for Manipulators with Rotary Joints 199

Figure 4 A 2R Arrangement Used for Orienting Vectors in Space

It is understood that the vector f

z , can be expressed with respect to the x y zˆ ˆ ˆ4 4 4-frame in terms of local

coordinates (n , 4o m and 4o l ), where 4o n and 4o m are unknowns to be worked 4o

out and l 4o is numerically obtained from = = f D

z by an angle θ5ˆz may be ex-5

pressed in the following form,

where c and 5 s stand for 5 cosθ5and sinθ5 respectively

A property of rotation about zˆ4 may be stated as,

This last expression (28) is a linear equation in s and 5 c This equation may 5

be re-expressed in a polynomial form as follows,

where t is the tangent of half θ5 and b j is the coefficient of the jth power term.

It could be concluded from equation (29), which is a second degree

polyno-mial, that the number of the wrist configurations, N wrist, which correspond to

the required orientation of f

From the analysis presented in this and the previous sections, it can be

con-cluded that the maximum number of configurations of a spherical-wrist

ma-nipulator structure which correspond to any given position and orientation at

the end-effector is eight The actual number of configurations, N, is calculated

by,

In spherical-wrist manipulators, each arm configuration corresponds to two

possible wrist configurations as indicated by equation (31)

7 Completing the Full Pose

Once the first five joints on the manipulator structure have performed

con-secutive rotations (θizˆ , wherei i=1, 2, 5) to place the sixth joint axis at its

desired position and orientation, one final rotation (θ6ˆz ), will be performed to 6

align any side axis on the end-effector with its desired direction The term

“side axis” here refers to any axis, on the end-effector Cartesian frame, whose

direction is influenced by rotations performed about ˆz This final part of the 6

inverse kinematic procedure is a straight forward application of the model

presented in equations (3) to (6) to calculate the angle of rotation However, it

worth noting here that this final step of the analysis is preceded by a direct

ki-nematic procedure to calculate the updated direction of the side axis after five

consecutive rotations, θ5zˆ5,θ4zˆ4,θ3zˆ3,θ2zˆ2andθ1 1zˆ ,have been performed

Inverse Position Procedure for Manipulators with Rotary Joints 201

8 The Inverse Solution Procedure

Figure (5) depicts a flow chart that has been designed to explain the procedure

proposed here for the inverse position analysis of manipulators For

spherical-wrist manipulators, the procedure produces eight sets of solutions in a

non-iterative fashion However, for calibrated robotic structures, the eight

solu-tions are obtained in a simple iterative approach which does not involve any

Jacobian matrix computations By virtue of the concepts presented, the

vari-ous solutions may be calculated simultanevari-ously if parallel computing facilities

are available

In the present approach, the arm is assigned the task of positioning any point

on the sixth joint-axis at its required spatial location The closest point on the

sixth-joint axis to the fifth joint-axis may be conveniently selected for this

pur-pose This point will be referred to in the following discussion as p0i

The four joint-displacement solutions which correspond to this positioning task are

therefore obtained using the models presented above and saved in four

three-element vectors, vj , where j=1,2,3 and 4.

At arm configuration number j, the wrist joints align the sixth joint-axis with

its required final orientation, as previously described, and the two

correspond-ing solutions are accordcorrespond-ingly obtained and saved in a pair of two-elements

vectors, wjk , where k may assume the values of 1 or 2 To this end, a set of

eight joint-displacement solutions have been obtained If the robot was of the

spherical-wrist type these solutions should accurately represent the required

joint-displacements and no iterations would be required

Calibrated robots, however, are not likely to have their last three joint-axes

in-tersecting at a common point (i.e the spherical-wrist property is lost), the

mo-tions performed by the wrist joints will displace the point which was

previ-ously positioned by the arm to eight new locations, p0jk, corresponding to the

wrist solutions obtained

At location number jk, the instantaneous position vector, p0jk, of the displaced

point may be calculated, using a suitable direct kinematic procedure, and

compared to the required position vector 0n

p where the net radial error, e jk, is calculated as follows,

≤

Read and Write Data

Perform Inverse Position Analysis (IPA) of the Arm to Get 4 Solutions

IPA of Wrist IPA of Wrist IPA of Wrist IPA of Wrist

m=1

IPA of arm to save the solution with minimum norm.

IPA of wrist to save the solution with minimum norm.

e ε

Get pjk

Compare to the required position and work out the error e.

No Yes

m

Stop

Figure 5 Inverse Position Analysis of Robots Using Elementary Motions

If the calculated value for e jk does not fall within an allowable error zone, the

calculations proceed such that at iteration number m, the arm sets out from the most updated configuration number jk(m-1) to position point pm-1 jk in the re-quired location, pn The four solutions obtained may be stored in four three-

Inverse Position Procedure for Manipulators with Rotary Joints 203

element vectors whose norms are subsequently calculated and compared

jk

v , may be saved in the memory and the other solutions would be discarded This vector is re-

ferred to here as the arm elementary-motions vector because it contains

frac-tional quantities of elementary joint-displacements

The two corresponding wrist solutions may then be obtained and stored in a

pair of two-element vectors whose norms will also be calculated and

com-pared The vector with minimum norm, wm jk, is subsequently saved while the

other vector may be disposed of In the current context, m

jk

w is designated as

the wrist elementary-motions vector because it contains small values of

joint-displacements

The new displaced location of the positioned point may then be calculated and

compared with the required location as per equation (32) When the radial

er-ror is small enough, the final joint-displacement vector, vn jk, of the arm group

which corresponds to solution number jk may be calculated as follows,

Once the jk-solution for the first five joint-displacements has been obtained,

the corresponding displacement of the last joint may simply be calculated

The iterative technique presented here utilises the physical kinematic

behav-iour of manipulator joints and therefore fast and singularity-proof

conver-gence may be assured The technique does not require initial guesses

intro-duced into the model

In the next section a numerical example is given where the inverse position

so-lutions will be produced for a PUMA-type robot of both calibrated and ideal

structures

9 Numerical Example

A PUMA-like manipulator with six revolute joints is selected for the example The dimensions of the spherical-wrist structure of the manipulator are given in Table (1) The dimensions of the non-spherical-wrist version of the same ma-nipulator are similar to those given in Table (1) except for the locations of the fourth and fifth joint-axes which were displaced to (-128.0, 818.51 and 205.04 mm) and (-130.5, 802.0 and 180.4 mm) respectively

Direction Cosines of Joint-axes Axes Locations (mm)

z3 -0.9961947 0.05233595 0.0696266 -68.0 438.0 195.0 z4 0.02233595 -0.9993908 0.02681566 -130.5 808.5 177.0 z5 0.99975050 -0.0223359 0.00009744 -130.5 808.5 177.0 z6 0.02489949 0.9996253 0.00012081 -130.0 808.5 177.0 Table 1 Cartesian Dimensions of a Spherical-wrist Manipulator

In both cases, the initial and final locations of the Tool Centre Point (TCP) of the end-effector are given with respect to the base coordinates as, -120.54, 1208.36 and 175.095 and, –400.0, –400.0 and 1009.0 mm respectively The initial and final orientations of the end-effector are given in terms of an Euler ZYZ-system as: 88.5733, 89.9604 and 89.722 and, 120.0, –20.0 and 150.0 degrees re-spectively

The models proposed in this paper were used to calculate the inverse position solutions for both the spherical-wrist and general manipulator structures and the results are displayed in Tables (2) and (3) respectively The angular dis-placements given in these tables are in degrees

Sol No 6 104.39 -72.90 64.57 -156.27 -89.56 177.24

Sol No 8 117.07 -11.05 -72.30 -120.58 -33.16 -122.18Table 2 Inverse Position Solutions for the Spherical-wrist Robot

Inverse Position Procedure for Manipulators with Rotary Joints 205

1

Sol No 1 -39.20 -162.93 66.79 86.5 -36.82 -125.19 Sol No 2 -30.78 -162.92 70.95 -73.54 32.47 68.51 Sol No 3 -48.12 -106.90 -68.18 19.18 -83.65 156.37 Sol No 4 -46.46 -103.38 -65.35 -158.64 72.85 -17.25 Sol No 5 105.11 -73.14 67.49 24.37 87.28 0.42 Sol No 6 103.60 -70.98 72.08 -154.89 -98.19 173.67 Sol No 7 120.24 -11.00 -68.57 65.31 29.99 60.59 Sol No 8 114.00 -11.34 -62.96 -133.30 -37.94 -134.91 Table 3 Inverse Position Solutions for the Non-spherical-wrist Robot

The solutions obtained for the spherical-wrist manipulator did not involve erations at all However, a maximum of 4 iterations were used for the non-spherical-wrist manipulator In most cases the number of iterations was 2 ex-cept for the second and first solutions were this number was 3 and 4 respec-tively This demonstrates the numerical efficiency of the proposed models

it-10 Conclusions

The work presented in this paper introduces a technique for inverse position analysis of revolute-joint manipulators The analysis developed results in simplified solutions for both the arm and the wrist subassemblies These solu-tions are obtained in form of polynomials whose coefficients can be simply calculated for a given manipulator structure The technique can be used to ob-tain inverse kinematic solutions for both spherical-wrist and calibrated ma-nipulator structures

The technique results in obtaining multiple sets of the joint-motor ments which correspond to a given pose at the end-effector This enables the trajectory designer to the select the joint-trajectory which best fits a desired manipulator task

displace-Appendix A

To relate a pair of successive axes on a manipulator structure, the direction

co-sines of the two axes are given (with respect to a Cartesian base frame),

to-gether with a position vector describing a point on each axis These spatial

particulars are defined in Figure (A) as zˆi and p0i for the joint axis number i

and zˆi+1 and p0(i+1) for joint-axis number i+1 The procedure kicks off by

cal-culating the common normal, xi+1 as follows;

Figure A Relating Successive Axes with a Common Normal

The shortest distance, a i+1, separating the two axes is calculated as follows;

+1 = 0(+1)− 0 Dˆ +1

Inverse Position Procedure for Manipulators with Rotary Joints 207

The intersection of xˆi+1 with zˆi is defined by a position vector pi, which is

which is subject to the condition, if (zˆi+1×xˆi+1)Dzˆi =0 then b i =0 (A.7)

The intersection of xˆi+1 with zˆi+1 is defined by a position vector pi i(+1), which is

Any two spatial circles intersect at two points if, and only if, their axes lie in

one and the same plane

Proof:

Figure (B) depicts two spatial circles, C1 and C2, and their axes, ˆz and 1 ˆz re-2

spectively The circles intersect one another at two points, s and 1 s To prove 2

that zˆ1 and zˆ2 must lie in one and the same plane, the centres of the two

cir-cles, Pc1 and Pc2, are connected to the point, s3, which divides the line s s1 2 into

two equal parts

• s s1 2 lies in a plane perpendicular to ˆz and therefore 1 s s1 2 is perpendicular to

1

ˆz

• From planar geometry, s s1 2 is perpendicular to the line Pc1 3s .

• Therefore s s1 2 is perpendicular to the plane which contains the two ting lines, ˆz and 1 Pc1 3s Let this plane be referred to as PN1

intersec-• Similarly, it could be established that s s1 2 is also perpendicular to the plane which contains the two intersecting lines, z2 and Pc2 3s This plane may be referred to as PN2

• A general conclusion may now be drawn that, PN1 is parallel to PN2

• However, PN1 and PN2 share one common point, s3

• Therefore, the two planes coincide and zˆ1, Pc1 3s , zˆ2 and Pc2 3s must all lie

in one and the same plane

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to provide a kinematic constraint to eliminate an undesired motion of the effector Manipulators in which the cables have variable lengths are usually called cable-driven or wire-driven manipulators.

end-Cable-based manipulators posses several advantages over conventional rial/parallel link manipulators including:

se-1 Large workspace: An active winch can provide a large range of length change on the cables at a low cost This facilitates building manipulators for very large working spaces which cannot be obtained by other robots

2 Low inertia: Materials provide their highest strength-to-mass ratio when they are under tensile loading Using cables, which can be only in tension, maximizes the use of material strength and therefore reduces the mass and inertia of the manipulator Low inertia is desirable in many applicati-ons including high speed/acceleration robotics

3 Simplicity in structure: Cables simplify the robot structure by utilizing bending flexibility as kinematic joints and reducing the fabrication cost by minimizing the machining process

4 Reconfigurability and transportability: Winch assemblies can be simply located to reconfigure and adjust the workspace of a cable-driven manipu-lator The ease of assembly/disassembly of these manipulators also facili-tates their transportation and quick setup

re-5 Fully remote actuation: Using a fully cable-driven manipulator, all the tuators and sensitive parts are located away from the end-effector and the actual working area Such manipulators best suit harsh or hazardous envi-ronments

It should be also noted that using cable structures in robot manipulators is companied by theoretical and technical difficulties The unilateral force of ca-bles complicates the workspace, kinematics and dynamics analysis The con-straint of tensile force in all cables should be incorporated into the design and control procedure otherwise, the manipulator will collapse Also, the low stiff-ness of the cables compared to rigid links may result in undesired vibrations requiring compensation by a proper control scheme

ac-As it was mentioned before, maintaining positive tension (tensile force) in all the cables is an essential requirement for the rigidity of a cable-based manipu-lator and hence, this property should be studied thoroughly before the cable-based manipulator can be used in any real application In other words, a cable-based manipulator can be treated as a rigid link manipulator only if all the ca-bles are in tension As a result, most of the researchers’ efforts on this category

of robot manipulators have been spent on analyzing and proving the rigidity

of the cable-based structures

The general problem of rigidity in cable-based manipulators has been studied

in the literature using different approaches and terminologies such as lable workspace (Verhoeven & Hiller, 2000), dynamic workspace (Barette & Gosselin, 2005), wrench closure (Gouttefarde & Gosselin, 2006), manipulability ( Gallina & Rosati 2002), fully constraint configuration (Roberts et al 1998) and tensionability (Landsberger & Shanmugasundram, 1992) General formulation

control-of this problem can be found in the works by (Ming & Higuchi 1994), koro et al., 1996), and (Verhoeven et al., 1998) They showed that for the rigid-ity of a cable-based manipulator, it is necessary but not sufficient to have either actuation redundancy or separate external loading sources to produce tension

(Tado-in all cables M(Tado-ing (M(Tado-ing & Higuchi 1994a,b) calls the first group Completely Restrained Positioning Mechanisms, CRPM, in which all the cables can be made taut with no external load while in an IRPM (Incompletely Restrained Positioning Mechanism), the manipulator cannot maintain its own rigidity and hence needs external load to make all cables taut

The useful workspace of a cable-based manipulator is a subset of its cal workspace in which the manipulator can be rigidified (either by actuation redundancy or external loading) Determination of this workspace is the most essential step in the design and operation of a cable-based manipulator and is usually done by numerical search after the synthesis of the manipulator is done Examples of this approach can be found in (Kawamura et al., 1995; Fer-raresi, 2004; Ogahara, 2003; So-Ryeok et al., 2005a,b; Pusey et al., 2004) In this approach, if the workspace found through the search does not satisfy the de-sign requirements, the synthesis of the manipulator and the workspace deter-mination should be repeated As a result and in order to avoid trial and error

geometri-in the design, it is desired to have cable-based manipulators that can be fied everywhere in their geometrical workspace or at least their workspace can

rigidi-be analytically expressed In this regard, a geometrical explanation for the

Cable-based Robot Manipulators with Translational Degrees of Freedom 213 workspace of a cable crane has been found (Landsberger & Shanmugasun-dram, 1992) which is an IRPM An analytical study for the boundaries of the workspace in planar cable-based manipulators is also performed in (Barette & Gosselin, 2005), (Gouttefarde & Gosselin, 2006) and (Stump & Kumar, 2006)

In this article, a series of cable-based manipulators with translational motion (Behzadipour, 2005) is studied with focus on their rigidity study In these de-signs, cables are used to drive the end-effector as well as to eliminate its rota-tion by proper kinematic constraints The significance of these new manipula-tors is that their rigidity can be guaranteed everywhere in their geometrical workspace by a certain set of conditions enforced on the geometry of the ma-nipulator This will be proved in details for each manipulator and the condi-tions will be derived By incorporating these conditions into the design and control, the cables will be always taut and hence the cable-based manipulator can be treated as its rigid link counterpart

In Section 2, the general structure of these manipulators will be presented and the critical concepts of rigidity and tensionability will be defined In Section 3,

a theorem is given to simplify the study of tensionability in these tors In Sections 4 and 5, two spatial cable-based manipulators are introduced and their rigidity are proved In Section 6, two planar manipulators with trans-lational motion are presented and their rigidity are thoroughly studied

manipula-2 General Structure and Definitions

The general configuration of the cable-based manipulators studied in this per is shown in Fig 1

pa-The four main elements of these manipulators are:

1 Base: The fixed part of the manipulator to which the global system of

coordinate OXYZ is attached

2 End-effector: The moving body which carries the moving frame O′X ′Y′Z′

3 Cables: The flexible tendon elements with negligible mass and diameter

connected from one end to the end-effector at points P i (i=1,2, ,m) and

pulled from the other end at Q i The pulling actuator produces tension τi

inside the cable and can be simply a winch which pulls and winds the

cable or a separate mechanism that moves the cable's end (Q i) without changing its length Unit vectors uˆi (i=1,2, ,m) determine the direction of the cables and point towards the base Depending on the structure of the manipulator, there may be some extra pulleys to guide the cables The

number of cables, m, is equal to the dimension of the motion space of the end-effector Therefore, m is three and six for planar and spatial mecha-

nisms, respectively

Figure 1 General configuration of the cable-based manipulators studied in this paper

4 Spine: The element that produces a force between the base and the

end-effector in order to keep all the cables in tension The spine can be an

acti-ve element which generates a desired force It can be also a passiacti-ve ment such as a cylinder energized by compressed air or a compressive spring designed properly to provide the sufficient force required to main-tain tension in the cables The direction of the spine is shown by unit vec-

ele-tor wˆ pointing towards the end-effector

For any cable-based manipulator, an equivalent rigid link counterpart can be found by replacing each cable by a rigid link and ball-and-socket joints at the ends If the cable has a variable length, then a cylindrical element should be used to represent the cable in the rigid link manipulator This analogy is valid

as long as the cable-based manipulator is rigid according to the following nition:

defi-Rigidity: A cable-based manipulator is rigid at a certain pose with

re-spect to a given external load (including dynamic loads) and spine force if and only if all cables are in tension,

m i

i ≥0 =1,2, ,

τ A positive τi is considered as a tensile force in the cable

Cable-based Robot Manipulators with Translational Degrees of Freedom 215

It should be noted that the rigidity of a cable-based manipulator depends on the external load and therefore, dynamic forces should be also considered when the rigidity is evaluated As a result, rigidity is not a property of the ge-ometry only The rigidity analysis requires the motion, inertia and all exter-nally applied forces to be considered which complicates the process To over-come this problem, another property called tensionability is defined and used which only depends on the geometry and expresses the potential of the ma-nipulator for being rigid

Tensionability: A cable-based manipulator is called tensionable at a given

pose if and only if for any arbitrary external load there ists a finite spine force and a set of finite cable tensions to make the manipulator rigid

ex-Note that if a manipulator is tensionable and there is enough tensioning force available (by the spine and the cables), then the manipulator will be rigid un-der any external loading In other words, tensionability and large enough ten-sioning force together provide a sufficient condition for the rigidity The con-verse is that a manipulator may be rigid under a certain condition but not tensionable

It is important to note that both rigidity and tensionability deal with the tence of the static equilibrium condition for the manipulator in which all the cables are in tension and hence, the manipulator does not collapse However, they do not explain the nature of the equilibrium Considering the stiffness of the manipulator, it may be rigid (meaning that it is in static equilibrium with all cables in tension) although the equilibrium might be an unstable one which implies that any small disturbance on the end-effector results in the collapse of the manipulator It is known that the stability of the manipulator from the stiffness point of view is not specific to cable-based manipulators; however, it

exis-is shown in (Behzadipour & Khajepour, 2006) that the cable tensions may have

a significant effect on the stiffness and even destabilization of the manipulator

3 Tensionability

The goal of this section is to introduce an approach for the evaluation of sionability in a cable-based manipulator According to the definition, the ten-sionability of a manipulator must be evaluated for any arbitrary external load

ten-In the following, a theorem is introduced which gives a sufficient condition for the manipulator to be tensionable

The core idea of this theorem is to show that if positive tension (tensile force) can be generated in all the cables to any desired extent while the static equilib-

rium is satisfied in the absence of the external loads, then the manipulator can

be rigidified under any arbitrary external load by having enough pretension in

the cables

Theorem 1

A kinematically non-singular configuration of a cable-based manipulator is

tensionable if for an arbitrary positive spine force Fs (compressive force), the

static equilibrium equations of the manipulator have a solution with all

posi-tive cable tensions τi‘s

0 F

This theorem simply states that if the manipulator can stay rigid and statically

balanced under an arbitrary compressive spine force, it is tensionable and thus

can stay rigid for any external force and torque by choosing a large enough

spine force

Proof:

For the proof, it will be shown that such a manipulator can be made rigid for

any arbitrary external load The balance of forces for an arbitrary external force

Fe applied at O′ and moment M e is:

0 F F

The above equations have a set of nontrivial solutions forτi's since the

manipu-lator is assumed to be kinematically non-singular Since the above set of

equa-tions is linear w.r.t τi's, superposition can be applied to obtain the following

two sets of equations:

e i

i e i i e

¦

i

i s i i s

Cable-based Robot Manipulators with Translational Degrees of Freedom 217

i s

)

(

i i

s i i

i

Therefore, by increasing the spine force, the rigidity can be obtained and

hence, the manipulator is tensionable

The above theorem gives a sufficient condition for tensionability meaning that

there might be situations in which the spine force cannot produce tension in all

cables but the manipulator can be still rigidified In those cases, sources other

than spine may be used to generate tension in cables An example of such

cases will be studied in Section 5.1

As a result from theorem 1, to evaluate the tensionability, instead of dealing

with external load on the end-effector, we only need to show that the static

equilibrium of the end-effector for an arbitrary spine force can be obtained by

tensile forces in all of the cables This will ensure that the manipulator is

ten-sionable and thus can theoretically stand any external force and moment at the

end-effector By “theoretically” we mean that the required spine force and

ca-ble tensions are finite, although these forces may not be feasica-ble due to the

practical constraints The above approach is used later in this paper to evaluate

the tensionability of the new cable-based manipulators

In the rest of this paper, some new designs of reduced DoF1 cable-based

ma-nipulators are introduced The target application of these mama-nipulators is

high-speed pick-and-place operations in which, small objects (less than 1kg)

are moved with high speeds (more than 100 cycles per minute) High speed

and acceleration requires low inertia which makes cable-based manipulators

potential designs However, most of the current spatial cable-based

manipula-tors have 6 DoF while in pick-and-place applications, three translational axes

of motion with a possible rotational DoF for reorienting the object are

suffi-cient In the designs presented in this work, cables are used to constrain the

ro-tational motion of the end-effector in order to provide a pure translational

mo-1 DoF: Degree of Freedom

tion A more complete set of these designs can be found in (Khajepour et al.,

2003) One of these designs, DeltaBot, has been prototyped at the University of

Waterloo (Dekker et al 2006) It can perform up to 150 cycles/minute of dard pick-and-place on small objects (less than 500gr)

stan-4 BetaBot

In BetaBot, shown in Fig 2, the upper triangle is the end-effector and the

bot-tom one is the base Three pairs of parallel cables are attached to the effector and wound by three winches after passing through guide holes on the winch frames The winches are attached to the base Each pair of cables forms

end-a pend-arend-allelogrend-am such end-as ABCD end-as shown in the send-ame figure It is known thend-at

(Clavel, 1991), three parallelograms can eliminate the rotational motion of the end-effector The spine is connected to the end-effector and base using two ball-and-sockets or one universal joint and one ball-and-socket Therefore, the spine imposes no kinematic constraint on the end-effector

The equivalent rigid link manipulator for BetaBot is obtained by replacing each

cable with a slider with two ball-and-sockets at the ends In this equivalent manipulator, there are 13 bodies, 12 ball-and-socket and 6 prismatic joints The Gruebler equation gives 13×6 − 12×3 − 6×5 = 12 degrees of freedom There are

6 trivial DoF's due to the twist of the sliders and there is also one constraint on each pair of sliders which forces their displacements to be the same (because each pair of cables is wound using a single winch) Therefore, the end-effector has 12 − 6 − 3 = 3 DoF's which are translational

Since the size of the end-effector plays no role in the kinematics of BetaBot, the

end-effector can be shrunk to a point with three winches moving towards the

center of the base accordingly As a result, the kinematics of BetaBot becomes

identical to that of a tripod (Mianowski & Nazarczuk, 1990), or the Tsai nipulator (Tsai, 1996)

ma-The geometrical workspace of BetaBot is only limited by the maximum and

minimum lengths of the spine assuming that there are no limitations on the cables' lengths and therefore, the shape of the workspace is a half sphere above the base whose radius is determined by the maximum length of the spine It is clear that there is no possibility of any interference between the cables because

of the non-rotating motion of the end-effector

Cable-based Robot Manipulators with Translational Degrees of Freedom 219

Figure 2 BetaBot: a 3 DoF cable-based manipulator with pure translational motion

4.1 Tensionability of BetaBot

BetaBot is tensionable everywhere in its workspace providing that certain ditions are enforced on the geometry of the manipulator as illustrated in Fig 3:

con-Condition 1. End-effector has a triangular shape as shown in Fig 3 Each

pair of the parallel cables is attached to one edge of the gle,

trian-Condition 2 The guide holes on the winch frames are all on the same

plane and form a triangle called Base triangle This triangle is similar (and parallel) to the triangle of the end-effector but larger As a result, the end-effector along with the cables form a convex region or a polyhedral

Condition 3. Any two cables never become in-line

Condition 4. The connection points of the spine, O and O′ in Fig 3, are

on the base and end-effector triangles, respectively and have the same trilinear coordinates2 A direct result is that the spine never intersects with the faces of the polyhedral of Condition 2 (even if the spine and cables are extended from the base side)

2 The ratio between their distances from the triangle vertices are the same

Figure 3 The prefered geometry of BetaBot which guarantees its tensionability

To prove the tensionability, we use the theorem given in the last section

Therefore, for BetaBot to be tensionable, an arbitrary positive spine force

should be balanced by positive cable tensions For the proof, a geometrical

ap-proach is employed which shows that the solution of static equilibrium

equa-tions consists of a set of positive tensions Since the proof is lengthy, it is

pre-sented by four lemmas and one theorem

Lemma 1.

If BetaBot meets the above mentioned conditions, then the three planes each of

which formed by one pair of parallel cables intersect at one point such as R

(see Fig 4 )

Proof.

In Fig 4, consider the plane that includes P1, P2, B1 and B 2 : If B1P1 and B2P2 are

extended, they meet at a point called R and form triangle ¨B 1 RB 2 (otherwise

they will be parallel which implies the end-effector and the base triangle are

equal contradicting Condition 2) In this triangle we have:

2 1

2 1 2

2 2

1

2

1

B B

P P RB

RP B

B

P

Cable-based Robot Manipulators with Translational Degrees of Freedom 221

Figure 4 The three planes formed by parallel cables and the spine meet at a point

called R.

Now, consider the second plane that includes P2, P3, B2 and B3: If B2P2 and B3P3

are extended, they meet at a point called R′ Note that R′ and R are both on

the line of P 2 B 2 In triangle ¨B 2 R ′B 3 we have:

3 2

3 2 2

2 3

2

3

2

B B

P P B R

P R B

Considering that R and R′ are on the same line, Eq (11) states that R and R′

are coincident and a similar reasoning shows that the third plane also passes

Assume point R, as given in Fig 4, is connected to O′ and extended to intersect

the base at point O ′′ (not shown in the figure) It is needed to show that O ′′

co-incides with O This is shown by the following relations:

1 1 1

1 1

1

1:

RB

RP B O

P O B

O P

1 2

1 2 1

2

RB

RP RB

RP B

B P

2 2

2

2:

RB

RP B

O

P O B

O P O

2 1

1

1

1

B O

P O RB

RP RB

3 1

1

1

1

B O

P O RB

RP RB

2

1

1

B O

P O B

1

OB

P O OB